Having taken holy orders, he was for some years tutor to Lords Charles and Robert Spencer, younger sons of the second Duke of Marlborough. On 21 September 1761, he was preferred to the fifth prebendal stall in the church of Durham, and in April 1763, to a canonry at Christ Church, Oxford. On 1 July 1764, he took the degrees of B.D. and D.D. On 19 September 1771, he was made dean of Canterbury, and on 10 February 1775, bishop of Bangor.[3]

On the death of Archbishop Frederick Cornwallis, he was translated to the see of Canterbury, 26 April 1783,[4] on the joint recommendation of Bishops Robert Lowth and Richard Hurd, both of whom had declined the primacy. Though not a great ecclesiastic, Moore was an amiable and worthy prelate, a competent administrator, and a promoter of the Sunday-school movement and of missionary enterprise. He appears to have dispensed his patronage with somewhat more than due regard to the interests of his own family.[3]

Moore married twice, first, a daughter of Robert Wright, chief justice of South Carolina ; secondly, on 23 January 1770, Catherine, daughter of Sir Robert Eden, bart., of West Auckland. He left issue.[3]